Thank you for your query,
Whenever cold is there, there are chances of getting nose bleed will be there.
When the bleeding occurs posteriorly children will swallow it it and because of stomach irritation
by blood they will develop vomitings.
If there is still bleeding from the nostril
, press and hold that side nostril till bleeding stops.
Consult your Pediatrician
for the examination and for the appropriate medication for cold and cough
Hope I have answered your query.